# Lesson 7

The Root of the Problem

• Let’s look at relationships between volumes, areas, and scale factors using graphs and situations.

### Problem 1

A solid with volume 8 cubic units is dilated by a scale factor of $$k$$ to obtain a solid with volume $$V$$ cubic units. Find the value of $$k$$ which results in an image with each given volume.

1. 216 cubic units
2. 1 cubic unit
3. 1,000 cubic units

### Problem 2

A solid has volume 7 cubic units. The equation $$k=\sqrt{\frac{V}{7}}$$ represents the scale factor of $$k$$ by which the solid must be dilated to obtain an image with volume $$V$$ cubic units. Select all points which are on the graph representing this equation.

A:

$$(0,0)$$

B:

$$(1,1)$$

C:

$$(1,7)$$

D:

$$(7,1)$$

E:

$$(14,2)$$

F:

$$(49,2)$$

G:

$$(56,2)$$

H:

$$(27,3)$$

### Problem 3

A solid with surface area 8 square units is dilated by a scale factor of $$k$$ to obtain a solid with surface area $$A$$ square units. Find the value of $$k$$ which leads to an image with each given surface area.

1. 512 square units
2. $$\frac{1}{2}$$ square unit
3. 8 square units

### Problem 4

It takes $$\frac18$$ of a roll of wrapping paper to completely cover all 6 sides of a small box that is shaped like a rectangular prism. The box has a volume of 10 cubic inches. Suppose the dimensions of the box are tripled.

1. How many rolls of wrapping paper will it take to cover all 6 sides of the new box?
2. What is the volume of the new box?
(From Unit 5, Lesson 6.)

### Problem 5

A solid with volume 8 cubic units is dilated by a scale factor of $$k$$. Find the volume of the image for each given value of $$k$$.

1. $$k=\frac{1}{2}$$
2. $$k=0.6$$
3. $$k=1$$
4. $$k=1.5$$
(From Unit 5, Lesson 6.)

### Problem 6

A figure has an area of 9 square units. The equation $$y=\sqrt{\frac{x}{9}}$$ represents the scale factor of $$y$$ by which the solid must be dilated to obtain an image with area of $$x$$ square units. Select all points which are on the graph representing this equation.

A:

$$(0,0)$$

B:

$$(1,1)$$

C:

$$(1,3)$$

D:

$$(3,1)$$

E:

$$(9,1)$$

F:

$$(9,3)$$

G:

$$(18,2)$$

H:

$$(36,2)$$

(From Unit 5, Lesson 5.)

### Problem 7

Noah edits the school newspaper. He is planning to print a photograph of a flyer for the upcoming school play. The original flyer has an area of 576 square inches. The picture Noah prints will be a dilation of the flyer using a scale factor of $$\frac14$$. What will be the area of the picture of the flyer in the newspaper?

(From Unit 5, Lesson 4.)

### Problem 8

Angle $$S$$ is 90 degrees and angle $$T$$ is 45 degrees. Side $$ST$$ is 3 feet. How long is side $$SU$$?

(From Unit 4, Lesson 6.)